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<title>Atlas software user guide -- Cartan subgroups</title>
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<h2>Cartan subgroups</h2>
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<i>Last updated: October 15, 2005</i>
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Let G be a connected complex reductive algebraic group, with a fixed 
<a href="innerclass.html">inner class</a> of real forms, and fix a real form 
in the class, 
<a href="realforms.html">represented</a> by an involution &theta;. Let K be the
fixed-point group for &theta; in G. The &#8220;cartan&#8221; command will 
output representatives of the various K-conjugacy classes of &theta;-stable
Cartan subgroups in G. (These are in natural bijection with the 
G(<b>R</b>)-conjugacy classes of Cartan subgroups of G(<b>R</b>)).
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It turns out that there is a rather nice connection between the classifications
of Cartan subgroups of the various real forms of G belonging to the fixed
inner class. For the quasisplit form, the Cartans are in (1,1) correspondence
with (twisted) W-conjugacy classes of twisted involutions in W (a twisted 
involution is an element w in W satisfying w.&delta;(w)=1, where &delta; is 
deduced from the involution of the Dynkin diagram defined by the inner class.) 
This set TwConj(W,&delta;) carries a natural poset structure, with a minimal 
element c<sub>min</sub> which is the fundamental Cartan. For any other real 
form &theta;, the Cartans of (G,&theta;) may be identified with the interval 
[c<sub>min</sub>,c<sub>max</sub>] in TwConj(W,&delta;), where c<sub>max</sub> 
corresponds to the most split Cartan in (G,&theta;).
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For each conjugacy class, the program will output the 
<a href="tori.html">type</a> of
the Cartan subgroup as a real algebraic torus; note that this is relatively
subtle, as it depends not only on the Lie type of G but also on the actual
covering group chosen: a complex torus factor might become a compact times
split one, or conversely (try for instance G = <b>SL</b>(2).<b>SL</b>(2),
and G<sub>1</sub> = G/D, where D is the diagonal subgroup in the center of G, 
both for the complex real form (which is an inner class all by itself.) For 
G(<b>R</b>), you get a complex torus, which makes sense because the group is 
just <b>SL</b>(2,<b>C</b>); whereas for G<sub>1</sub>(<b>R</b>), which is 
<b>SO</b>(3,1), you get a compact times split torus. For 
G<sub>2</sub> = G/Z(G), which has 
G<sub>2</sub>(<b>R</b>)=<b>PSL</b>(2,<b>C</b>), you again get a complex torus.)
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Then the program will output data that depend only on the restriction of 
&theta; to the Cartan (and therefore are the same for all the real forms 
sharing this Cartan): the real and imaginary root systems, and what I call, for
lack of a better name, the "complex factor", which is the complex root system
defined in [1], orthogonal to both the half-sums of positive real and of 
positive imaginary roots. From these data, as explained in [1], one can easily
deduce the group W(&theta;) of &theta;-fixed elements in W.
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Finally, the program will output the classification of the weak real forms of
G for which this Cartan is defined. This amounts to the classification of the
orbits of the Weyl group of the imaginary root system, in a set that carries
a simply transitive action of the component group of the torus <it>dual</it>
to the corresponding Cartan in the adjoint group of G (notice that the Cartan
classifications for G and its adjoint group are the same, even when G is 
reductive.) To obtain the classification of strong real forms for this Cartan,
(or more precisely, the various combinatorial types of packets of strong real
forms, as explained <a href="strongreal.html">here</a>), use the 
&#8220;strongreal&#8221; command.
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<td>[1]</td>
<td>
David A. Vogan, Jr., Irreducible characters of semisimple Lie groups IV. 
Character-multiplicity duality, <i>Duke Math. J.</i> <b>49</b> (1982), no. 4, 
pp. 943--1073.
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